Techniques for Finding
Derivatives
Lesson 4.1
Limitations of the Definition
Recall our use of the definition of the
derivative
lim
h 0
f ( x  h)  f ( x )
h
This worked OK for simple functions
f ( x)  2 x 2  3 x
Becomes unwieldy for other common
functions
• Higher degree polynomials
• Trig functions
2
Other Ways to Represent The
Derivative
Previous chapter
• For a function f(x), we used f '(x)
To show derivative taken with respect to a
variable
dy
• When y is a function of x
Dx y
dx
shows "derivative of y with respect to x"
Other representations
Dx  f ( x)
3
Constant Rule
Given f(x) = k
• A constant function
Then
f ( x  h)  f ( x )
f '( x)  lim
h 0
h
When we evaluate this we get
k k
0
h 0
h
f '( x)  lim
We conclude when f(x) = k
• f '(x) = 0
How does this fit with
our understanding
that the derivative is
the graph of the slope
values?
4
Power Rule
Consider f(x) = x3
f '( x)  lim
h 0
Use the definition to
determine the derivative.
f ( x  h)  f ( x )
h
Now let h → 0
• f(x) = x3
• f '(x) = 3x2
What pattern do you see?
5
Power Rule
For f(x) = xn
• With any real number n
Then
f '( x)  n  x
Multiply the function
by the exponent
n 1
Decrease the
exponent by 1
6
Constant Times A Function
What happens when we have a constant
times a function?
• Example
f ( x)  k  x n
The rule is Dx  k  f ( x)  k  f '( x)
So
f '( x)  k  n  x n1
7
Sum Or Difference Rule
Consider a function which is the sum of two other
functions
• Example :
f ( x )  h( x )  k ( x )
The derivative of f(x) is
f '( x)  h '( x)  k '( x)
The derivative of the sum is the sum of the
derivatives
8
Try It Out
Apply all these rules to take the derivatives
of the following functions.
x
y  3x  x 
12
3
2
f ( x)  2 x
x2  2
f ( x) 
x
2.5
 8x
0.5
6
y 4
x
9
Marginal Analysis
Economists use the word "marginal" to
refer to rates of change.
When we have a function which represents
• Cost
• Profit
• Demand
Then the marginal cost (or profit, or
demand) is given by the derivative
10
Marginal Analysis
When the sales of a product is a function of
time t = number of years
S (t )  100  100t
1
What is the rate of change or the marginal
sales function?
• What is the rate of change after 3 years?
• After 10 years?
11
Assignment
Lesson 4.1A
Page 248
Exercises 1 – 45 odd
Lesson 4.1B
Page 250
Exercises 51 – 73 odd
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